We give a criterion for the hypoellipticity of a class of differentialoperators of Hormander type$$L = 1/2sum_{i = 0}^n X_i^2 + X_0.$$Our criterion is weaker than the usual finite-type Hormander condition and allows for decencies of $L$ of exponential order on hypersurfaces in the ambient space. The methods of proof are probabilistic, being based upon an analysis of the regularity of the diffusion process driven by $L$. These methods are also used to study the Dirichlet problem associated with exponentially degenerate operators $L$.